On Polynomially Integrable Domains in Euclidean Spaces

Mark Agranovsky

doi:10.1007/978-3-319-70154-7_1

On Polynomially Integrable Domains in Euclidean Spaces

Mark Agranovsky

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

4 Scopus citations

Abstract

Let D be a bounded domain in ℝⁿ, with smooth boundary. Denote VD(ω,t),ω∈Sⁿ⁻¹,t∈ℝ, the Radon transform of the characteristic function χ_D of the domain D, i.e., the (n − 1)-dimensional volume of the intersection D with the hyperplane { x∈ ℝⁿ: < ω, x> = t}. If the domain D is an ellipsoid, then the function V_D is algebraic and if, in addition, the dimension n is odd, then V (ω, t) is a polynomial with respect to t. Whether odd-dimensional ellipsoids are the only bounded smooth domains with such a property? The article is devoted to partial verification and discussion of this question.

Original language	English
Title of host publication	Trends in Mathematics
Publisher	Springer International Publishing
Pages	1-21
Number of pages	21
Edition	9781493976584
DOIs	https://doi.org/10.1007/978-3-319-70154-7_1
State	Published - 2018

Publication series

Name	Trends in Mathematics
Number	9781493976584
ISSN (Print)	2297-0215
ISSN (Electronic)	2297-024X

Bibliographical note

Publisher Copyright:
© Springer International Publishing AG 2018.

Keywords

Cross-section
Ellipsoid
Fourier transform
Polynomial
Radon transform

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10.1007/978-3-319-70154-7_1

Cite this

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KW - Cross-section

KW - Ellipsoid

KW - Fourier transform

KW - Polynomial

KW - Radon transform

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On Polynomially Integrable Domains in Euclidean Spaces

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